We will do the \(\theta \)  integral numerically.
The potential energy felt by an electron neutrino is the opposite. In this case, there is no potential energy for the tau neutrino. Such flavor dependent matter potential could affect the splitting among (effective) neutrino masses and their oscillation probabilities [1]}, [2]}. The effective Hamiltonian due to new physics contribution to the matter potential is parametrized as
\(H_{\rm BSM} = \sum _{\alpha , \beta = e, \mu , \tau } \sqrt{2} G_F \epsilon _{\alpha \beta } (\bar{\nu }_\alpha \gamma ^0 P_L \nu _\beta ) n_e \ .\) 
